Domanda del libro di testo
Evaluate the integrals in Exercises 31–78.
33. ∫e^x sec²(e^x - 7)dx
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Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 7, Problema 7.PE.3
In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
3. y = (1/4)xe^(4x) - (1/16)e^(4x)
Guida verificata passo dopo passo1
Identify the function to differentiate: \(y = \frac{1}{4} x e^{4x} - \frac{1}{16} e^{4x}\).
Recognize that the derivative of \(y\) with respect to \(x\) requires using the product rule for the first term \(\frac{1}{4} x e^{4x}\) and the chain rule for the exponential terms.
Apply the product rule to the first term: if \(u = \frac{1}{4} x\) and \(v = e^{4x}\), then \(\frac{d}{dx}(uv) = u'v + uv'\). Compute \(u' = \frac{1}{4}\) and \(v' = 4 e^{4x}\) using the chain rule.
Differentiate the second term \(- \frac{1}{16} e^{4x}\) using the chain rule: \(\frac{d}{dx} e^{4x} = 4 e^{4x}\), so multiply by the constant \(-\frac{1}{16}\).
Combine the derivatives from both terms to write the full expression for \(\frac{dy}{dx}\) before simplifying.
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Derivative of Exponential Functions
The derivative of an exponential function with base e, such as e^(kx), is found by applying the chain rule. Specifically, d/dx[e^(kx)] = k * e^(kx), where k is a constant. This rule is essential for differentiating terms like e^(4x).
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04:50
Derivatives of General Exponential Functions
Product Rule
The product rule is used to differentiate functions that are products of two differentiable functions. If y = u(x)v(x), then y' = u'v + uv'. This rule applies to terms like (1/4)x * e^(4x), where both factors depend on x.
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05:18The Product Rule
Constant Multiple Rule
The constant multiple rule states that the derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function. For example, d/dx[c * f(x)] = c * f'(x). This simplifies differentiation of terms like (1/4)xe^(4x) and (1/16)e^(4x).
Video consigliato:
04:02The Power Rule
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